Consider the Bose-Hubbard model for a single species of boson in a square lattice,
\begin{equation}
H_{a}=-t\sum_{<ij>} a^{\dagger}_{i}a_{j}+U \sum_{i}a^{\dagger}_{i}a^{\dagger}_{i}a_{i}a_{i}-\mu\sum_{i}a^{\dagger}_{i}a_{i}
\end{equation}
This model has a phase transition from Mott insulator to superfluid at $T=0$.
Now I add a one more type of boson species, say $b$, and there is a interaction between intra species of boson at same site, then we have
\begin{equation}
H=H_{a}+H_{b}+V\sum_{i}a^{\dagger}_{i}b^{\dagger}_{i}b_{i}a_{i}
\end{equation}
What will be the phase diagram of the system for filling($f$) $1$ and filling $1/2$ per site of each species? (Is there any symmetry breaking?)
